US Cs The steering control mechanism of a service mo- bile robot is represented by the following closed- loop system, where L(3) GS) = s(s+3) ; H(S) = 1 s(8+3)(8 + 6) L(S) = Controller (a) (5 pts) Design a of P controller (L(S) = K, K > 0): Using the Control Systems Toolbox" of Matlab, draw the system's root locus (you may find the command sisotool handy). Find K numerically such as the actual system's overshoot (not the approximated one is as close as possible to 20%. You may find the Matlab tool sisotool useful to find this value. (b) (10 pts) Response analysis: Numerically, find the properties of the system's actual step response (the overshoot, peak time, settling time, and rise time). Include a step response graph clearly showing as many as the above computed quantities as possible. Compute the position, velocity, and acceleration static error constants, and deduce the steady state errors corresponding to a unit step, unit ramp, and unit parabolic inputs. For each type of input, include a graph showing the input, the system's response, and the error vs. time. Even in the presence of a P controller, this kind of system is often considered uncompensated, because the gain K can be naturally absorbed as part of the plant GS). (c) (15 pts) Annihilating the steady-state error: Replace the P controller with a PI com- pensator, Li(8), to cancel the steady state error of the uncompensated system and to maintain (as close as possible) the transient response criterion set in question (a). Set the controller's zero at one tenth of the real part of the dominant poles of the uncompensated system. Find the value of this zero and of the two controller gains (K and K2), where L1(8) = Gp1(8) = Ki + K2_K1(8 + Zc) where Zci = K2K1 Repeat question (b) for this compensated system. Compare the results with the performance of the uncompensated system